Local and Global Solutions

I am having great fun taking a Graph Theory class this semester. A graduate level math class, it focuses very much on the theory of graphs, which is remarkably different from the real-world networks I’ve been growing accustomed to.

(And for those of you playing at home, a “graph” and a “network” are basically the same thing, but “graph” is the math/theoretical term and “network” is the science/real-world term.)

In each class, we’re basically asked to prove or disprove properties of a given graph. This is harder than it sounds.

The hardest part, actually, is that I usually think I know the answer. There’s something about the functioning of networks – sorry, graphs – that generally seems intuitively clear. But even if I know the answer, I have no idea how to actually prove it. That’s where the fun comes in.

Now about a month into the semester, I’ve notice an interesting trend in my (flawed) approach to proofs. Asked to explain why a certain property cannot be true, I immediate argue why it couldn’t possibly happen at the local level.

If it’s not a problem at the local level, it ought not to be a problem at the global level.

That’s essentially every argument I’ve made so far.

And it’s not necesarily that I’m wrong about the local level, but – networks are fickle things and a localized approach runs the danger of aggregating into something unintended. That is – you can’t just aggregate the local to make inferences about the global.

Frankly, this is one of the reasons I’m studying network science. Networks are complex, dynamic models which can so easily be broken down and analyzed. To really understand what’s going on you need to appreciate the local and the global, and think more broadly about how the whole structure interacts.